Are $+\infty$ & $-\infty $ elements of the real number line?

Question

Can someone give an explanation/proof of whether these two numbers lie on the real number line?

Answer 1

They are not real numbers as such. They are part of the extended real number line $\bar{\Bbb R}=\Bbb R \cup\{\infty,-\infty\}$. These elements do not have the usual arithmetic with other real numbers and abide strict rules however they are useful in fields such as analysis. They provide an unlimited behavior to certain concepts such as limits and integrals.

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Let me add to why they are not in the real number line. The rationals are dense in $\Bbb R$ so every real number can be represented as the limit of a sequence of rational numbers. This can be done from both sides of any real number which poses obvious problems if 2 elements are the largest and the smallest etc.